Relevant Applications of Differential Algebra in Astrodynamics

Transcription

1 AstroNet-II International Final Conference Tossa de Mar June 2015 Relevant Applications of Differential Algebra in Astrodynamics Pierluigi Di Lizia in collaboration with: Roberto Armellin, Alessandro Morselli, Monica Valli, Alexander Wittig, Franco Bernelli Zazzera

2 2 Outline Differential Algebra (1 slide) Building blocks (BB): ODEs flow expansion and uncertainty propagation (BB1) Taylor expansion of parametric implicit equations (BB2) Taylor expansion of two-point boundary value problems (BB3) Applications: SST: Impact Probability Computation Optimal control High-order filters Automatic domain splitting Spacecraft reentry NEO motion propagation

3 3 Outline Differential Algebra (1 slide) Building blocks (BB): ODEs flow expansion and uncertainty propagation (BB1) Taylor expansion of parametric implicit equations (BB2) Taylor expansion of two-point boundary value problems (BB3) Applications: SST: Impact Probability Computation Optimal control High-order filters Automatic domain splitting Spacecraft reentry NEO motion propagation

4 4 Differential Algebra Differential Algebra (DA) is an automatic differentiation technique Algebra of real numbers Algebra of Taylor polynomials Unlike standard automatic differentiation tools, the analytic operations of differentiation and antiderivation are introduced DA can be easily implemented in a computer environment (COSY-Infinity - Berz&Makino 1998, DACE, Jet Transport, JACK) Given any sufficiently regular function f of, DA enables the computation of its Taylor expansion up to an arbitrary order v n

5 BB1: Uncertainty propagation low computational burden low accuracy computationally intensive high accuracy Can we find a compromise technique? We need a technique to: Improve accuracy of linearized models Reduce computational cost of classical Monte Carlo 5

6 BB1: Uncertainty propagation low computational burden low accuracy computationally intensive high accuracy Can we find a compromise technique? We need a technique to: Improve accuracy of linearized models Reduce computational cost of classical Monte Carlo Differential Algebra 6

7 7 BB1: Expansion of the Flow of Dynamics Given any dynamics Initialize initial conditions as a DA vector: Perform all operations of the integration scheme in the DA algebra E.g.: Euler s scheme:

8 8 BB1: Expansion of the Flow of Dynamics Given any dynamics Initialize initial conditions as a DA vector: Perform all operations of the integration scheme in the DA algebra E.g.: Euler s scheme:

9 9 BB1: Expansion of the Flow of Dynamics Given any dynamics Initialize initial conditions as a DA vector: Perform all operations of the integration scheme in the DA algebra E.g.: Euler s scheme: k-th order Taylor expansion of the solution at

10 10 BB1: Expansion of the Flow of Dynamics Given any dynamics Initialize initial conditions as a DA vector: Perform all operations of the integration scheme in the DA algebra E.g.: Euler s scheme:

11 11 BB1: Expansion of the Flow of Dynamics Given any dynamics Initialize initial conditions as a DA vector: Perform all operations of the integration scheme in the DA algebra E.g.: Euler s scheme:

12 12 BB1: Expansion of the Flow of Dynamics Given any dynamics Initialize initial conditions as a DA vector: Perform all operations of the integration scheme in the DA algebra E.g.: Euler s scheme: k-th order Taylor expansion of the solution at

13 13 BB1: Expansion of the Flow of Dynamics Given any dynamics Initialize initial conditions as a DA vector: Perform all operations of the integration scheme in the DA algebra E.g.: Euler s scheme: k-th order Taylor expansion of the solution at

14 14 BB1: Expansion of the Flow of Dynamics Example: 2-body dynamics Eccentricity: 0.5 Starting point: pericenter Integration scheme: Runge-Kutta (variable step, order 8) DA-based ODE flow expansion order: 6 Initial box: AU in x and 0.08 AU in y

15 BB1: Uncertainty Propagation Methods Linear covariance propagation: If all computations are performed to order 1, the final Taylor expansion coincides with the state-transition matrix Initial covariance can be propagated as: 15

16 BB1: Uncertainty Propagation Methods Linear covariance propagation: If all computations are performed to order 1, the final Taylor expansion coincides with the state-transition matrix Initial covariance can be propagated as: No need of variational equations! 16

17 17 BB1: Uncertainty Propagation Methods DA-based Monte Carlo (MC): Any pointwise integration can be replaced by the evaluation of the polynomial Saving in computational time w.r.t. classical MC

18 18 BB1: Uncertainty Propagation Methods DA-based Monte Carlo (MC): Standard MC DA-based MC Any pointwise integration can be replaced by the evaluation of the polynomial Note: The accuracy of Taylor expansion can be tuned with order The same polynomial can be used to map different uncertainties

19 19 BB1: Uncertainty Propagation Methods DA-based Monte Carlo (MC): Any pointwise integration can be replaced by the evaluation of the polynomial Note: The accuracy of Taylor expansion can be tuned with order The same polynomial can be used to map different uncertainties

20 20 BB1: Uncertainty Propagation Methods Polynomial bounder:

21 21 BB1: Uncertainty Propagation Methods Polynomial bounder: Bounds of Polynomial bounders can immediately estimate the range of

22 BB1: Uncertainty Propagation Methods Polynomial bounder: Bounds of Polynomial bounders can immediately estimate the range of Example: very useful to discard the occurrence of an event Impact of NEOs with Earth 22

23 23 BB2: Parametric Implicit Equations Kepler s equation: Given a reference solution for and, suppose and are affected by uncertainties and Find the solution of the resulting parametric eq.: Initialize,, and as DA variables and compute:

24 24 BB2: Parametric Implicit Equations Kepler s equation: Given a reference solution for and, suppose and are affected by uncertainties and Find the solution of the resulting parametric eq.: Initialize,, and as DA variables and compute: Invert the map and enforce constraint

25 25 BB2: Parametric Implicit Equations Kepler s equation: Nominal values:,, Expansion order: 6 solution accuracy

26 26 BB3: TPBVP solver Given the Lambert problem: with tof

27 27 BB3: TPBVP solver Given the Lambert problem: r f with tof Consider displaced initial/final positions (e.g., initial error or moving target) r 0 Find the initial velocity to fly from r 0 + r 0 to r f + r f in tof

28 28 BB3: TPBVP solver Given the Lambert problem: r f with tof Consider displaced initial/final positions (e.g., initial error or moving target) r 0 Find the initial velocity to fly from r 0 + r 0 to r f + r f in tof Compute the Taylor expansion of

29 29 BB3: TPBVP solver Given the Lambert problem: r f with tof Consider displaced initial/final positions (e.g., initial error or moving target) r 0 Find the initial velocity to fly from r 0 + r 0 to r f + r f in tof Compute the Taylor expansion of and invert the map: Expansion of Lambert s problem solution

30 30 BB3: TPBVP solver Example: constraint on final position r f =0 r f tof r 0

31 31 BB3: TPBVP solver Example: constraint on final position r f =0 Impose r f =0: r 0 tof

32 32 BB3: TPBVP solver Test case: Earth-Mars transfer (Mars Express) nominal

33 33 BB3: TPBVP solver Test case: Earth-Mars transfer (Mars Express) error box of 0.1 AU

34 34 BB3: TPBVP solver Test case: Earth-Mars transfer (Mars Express) No correction 5 th order correction

35 Applications SST: Impact Probability Computation (BB1&BB2) Optimal control (BB1&BB3) High-order filters (BB1) 35

36 36 SST: Impact Probability Computation Suppose no impact occurrence for nominal initial states x sat 1 0 t DCA x sat 2 0

37 37 SST: Impact Probability Computation Suppose no impact occurrence for nominal initial states Impact conditions may occur for x sat i 0 6= x sat i 0 at t DCA 6= t DCA x sat 1 0 x sat 2 0

38 38 SST: Impact Probability Computation Suppose no impact occurrence for nominal initial states Impact conditions may occur for x sat i 0 6= x sat i 0 at t DCA 6= t DCA DA can be used to compute the Taylor expansion of the distance of closest approach (DCA) w.r.t. initial states: x sat 1 0 x sat 2 0

39 39 SST: Impact Probability Computation Suppose no impact occurrence for nominal initial states Impact conditions may occur for x sat i 0 6= x sat i 0 DA can be used to compute the Taylor expansion of the distance of closest approach (DCA) w.r.t. initial states: Compute: [x sat i]=t sat x i ( x sat i 0, t f ) for i =1, 2 f f at t DCA 6= t DCA [d 2 ]=T d 2( x sat i 0, t f ) [@d 2 /@t f ]=T d 2( x sat i 0, t f ) x sat 1 0 x sat 2 0

40 40 SST: Impact Probability Computation Suppose no impact occurrence for nominal initial states Impact conditions may occur for x sat i 0 6= x sat i 0 at t DCA 6= t DCA DA can be used to compute the Taylor expansion of the distance of closest approach (DCA) w.r.t. initial states: Compute: [x sat i]=t sat x i ( x sat i 0, t f ) for i =1, 2 f f [d 2 ]=T d 2( x sat i 0, t f ) [@d 2 /@t f ]=T d 2( x sat i 0, t f ) Invert: [ t f ]=T d 2( x sat i 2 /@t f 2 /@t f =0 Impose : [t DCA 2 =0( x sat i 0 ) [d DCA ]=T DCA ( x sat i 0 ) x sat 1 0 x sat 2 0

41 SST: Impact Probability Computation [d DCA ]=T DCA ( x sat i 0 ) DA-based fast Monte Carlo simulations can be run on to compute # of samples with x sat 1 0 x sat 2 0 threshold The map T DCA ( x sat i 0 ) was used for: DA-based Monte Carlo (DAMC) DA-based Line Sampling (DALS) DA-based Subset Simulation (DASS) more efficient for low probabilities Different orbital regimes: LEO, MEO, GEO 41

42 42 SST: Impact Probability Computation Efficiency of LS and SS over standard Monte Carlo for low probabilities: Cumulative impact probability distribution for a close conjunction # of samples: 14000

43 43 SST: Impact Probability Computation Efficiency of LS and SS over standard Monte Carlo for low probabilities: Cumulative impact probability distribution for a close conjunction # of samples: No samples generated by standard MC below 5 m distance

44 SST: Impact Probability Computation Efficiency of DAMC, DASS, and DALS over standard MC Three test cases (Alfano, 2009) Test 5: linear relative motion ( P c = E-2 ) Test 6: boundary of linear relative motion ( P c = E-3 ) Test 7: nonlinear relative motion ( P c = E-4 ) * Intel Core i5 8Gb RAM 44

45 SST: Impact Probability Computation Efficiency of DAMC, DASS, and DALS over standard MC Three test cases (Alfano, 2009) Test 5: linear relative motion ( P c = E-2 ) Test 6: boundary of linear relative motion ( P c = E-3 ) Test 7: nonlinear relative motion ( P c = E-4 ) * Intel Core i5 8Gb RAM 45

46 SST: Impact Probability Computation Efficiency of DAMC, DASS, and DALS over standard MC Three test cases (Alfano, 2009) Test 5: linear relative motion ( P c = E-2 ) Test 6: boundary of linear relative motion ( P c = E-3 ) Test 7: nonlinear relative motion ( P c = E-4 ) * Intel Core i5 8Gb RAM 46

47 SST: Impact Probability Computation Efficiency of DAMC, DASS, and DALS over standard MC Three test cases (Alfano, 2009) Test 5: linear relative motion ( P c = E-2 ) Test 6: boundary of linear relative motion ( P c = E-3 ) Test 7: nonlinear relative motion ( P c = E-4 ) * Intel Core i5 8Gb RAM 47

48 48 Optimal Control Optimal low-thrust state-to-state transfer problem x f Continuous thrust: Dynamics: 8 ṙ = v >< v = >: ṁ = µ r 3 r + T max u m T u 2 [0, 1] max u I sp g 0 control term x 0 Minimize: J = T max I sp g 0 Z tf t 0 u dt minimize fuel consumption J = Z tf t 0 u 2 dt minimize energy

49 49 Optimal Control Control theory reduces optimal control problem to a TPBVP: Dynamics 8 ṙ = v v = µ r 3 r Tu v m v Tu >< ṁ = I sp g 0 r = µ 3 µ r v r 3 v r 5 v = r >: m = Tu m 2 v r Initial conditions: x(t 0 )=x 0, m(t 0 )=m 0 Final conditions: x(t f )=x f, m(t f )=0 8 >< u =0 if > 0 where u =1 if < 0 >: u 2 [0, 1] if =0, switching function I sp g 0 =1 m v m DA can be applied to expand the solution of the optimal control problem with respect to the initial and/or final state

50 50 Optimal Control Reference solution, Y [AU] X [AU] Thrust arc Coast arc Thrust magnitude Time

51 51 Optimal Control Reference solution, Thrust magnitude Time Uncertainty: Problem: DA-based solution: x 0 or x f x,m(t 0 )=T (x 0, x f ) x,m(t 0 )?

52 52 Optimal Control Algorithm: Initialize: [x 0 ]=x 0 + x 0 [ x 0,m 0 ]= x0,m 0 + x0,m 0 DA-based integration: [x f, x f,m f ]=T xf, f ( x 0, x 0,m 0 ) Extract: [x f ]=T xf ( x 0, x 0,m 0 )

53 53 Optimal Control Algorithm: Initialize: [x 0 ]=x 0 + x 0 [ x 0,m 0 ]= x0,m 0 + x0,m 0 DA-based integration: [x f, x f,m f ]=T xf, f ( x 0, x 0,m 0 ) Extract: [x f ]=T xf ( x 0, x 0,m 0 ) Invert: [ x 0,m 0 ]=T 0 ( x 0, x f ) Taylor expansion of the solution of the optimal control problem from x 0 + x 0 to x f + x f

54 54 Optimal Control: Lunar Landing Lunar landing: minimum energy, uncertain initial state Max pos. error: 1 km Max vel. error: 5 m/s no correction

55 55 Optimal Control: Lunar Landing Lunar landing: minimum energy, uncertain initial state Max pos. error: 1 km Max vel. error: 5 m/s 5 th order OCP expansion control correction profiles

56 56 Optimal Control: Earth-Mars Transfer Earth-Mars transfer: minimum energy, uncertain final state Departure date: 1214 MJD2000 km Transfer time: 2413 days Uncertainty: 0.1 AU Expansion order: x y [AU] 0 u [m/s 2 ] x [AU] epoch [MJD2000] x 10 5

57 57 Optimal Control: Earth-Mars Transfer Earth-Mars transfer: minimum energy, uncertain final state Departure date: 1214 MJD2000 km Transfer time: 2413 days Uncertainty: 0.1 AU Expansion order: x y [AU] 0 u [m/s 2 ] x [AU] epoch [MJD2000] x 10 5

58 58 Optimal Control: Earth-Mars Transfer Earth-Mars transfer: minimum energy, uncertain final state Accuracy analysis (macbook 1.1, 2GHz Intel Core Duo) 1st order 2nd order CPU time: s CPU time: s

59 59 Optimal Control: Earth-Mars Transfer Earth-Mars transfer: minimum energy, uncertain final state Accuracy analysis (macbook 1.1, 2GHz Intel Core Duo) 3rd order 4th order CPU time: s CPU time: s

60 Optimal Control Earth FG3 transfer: minimum fuel, uncertain initial state δr f r f Max pos. error: AU Expansion order: 4 r 0 + δr 0 r 0 reference 1996 FG 3 1 Thrust arc Coast arc Y [AU] Earth Thrust magnitude X [AU] * ESA/ACT Ariadna Study Time 60

61 Optimal Control Earth FG3 transfer: minimum fuel, uncertain initial state δr f r f Max pos. error: AU Expansion order: 4 r 0 + δr 0 r 0 reference Y [AU] 1996 FG Earth Thrust arc Coast arc X [AU] * ESA/ACT Ariadna Study 61

62 Optimal Control Earth FG3 transfer: minimum fuel, uncertain initial state δr f r f Max pos. error: AU Expansion order: 4 r 0 + δr 0 r 0 reference 1996 FG Earth Thrust arc Coast arc Y [AU] X [AU] * ESA/ACT Ariadna Study 62

63 Optimal Control Earth FG3 transfer: minimum fuel, uncertain initial state δr f r f Max pos. error: AU Expansion order: 4 r 0 + δr 0 r 0 reference 1996 FG Earth Thrust arc Coast arc Y [AU] X [AU] * ESA/ACT Ariadna Study 63

64 Optimal Control Earth FG3 transfer: minimum fuel, uncertain initial state δr f r f Max pos. error: AU Expansion order: 4 r 0 + δr 0 r 0 reference 1996 FG Earth Thrust arc Coast arc Y [AU] X [AU] * ESA/ACT Ariadna Study 64

65 High-order Kalman Filters Given the discrete system model: [Based on Park & Scheeres, 2007] Suppose to have the state x k, with mean m + k and covariance P + k General filtering algorithm: Prediction: Update: 65

66 High-order Kalman Filters Given the discrete system model: [Based on Park & Scheeres, 2007] Suppose to have the state x k, with mean m + k and covariance P + k General filtering algorithm: classical approach: linearization Prediction: Update: 66

67 High-order Kalman Filters DA provides high-order polynomials for x k+1 = T k,k+1 ( x k ) z k+1 = T hk+1 ( x k+1 ) Expectations can then be computed on polynomials. E.g.: m k+1 = mx p=1 1 p! 1... p k,k+1 E( x k 1... x p k ) coefficients of the polynomials where E( x 1 k... x p k ) is computed using Isserlis formula Prediction step is fully nonlinear: Faster convergence Possible reduction of minimum measurements acquisition frequency to have convergence 67

68 High-order Kalman Filters Example: 2-Body Problem State variables: S/C position and velocity Uncertainty on initial conditions (10% off from true initial state) and no process noise Nonlinear measurements with realistic measurement noise (radial position of the S/C wrt the Earth and the LOS direction to Earth) DAHNEKF 1 DAHNEKF 2 DAHNEKF DAHNEKF 1 DAHNEKF 2 DAHNEKF 3 ε r [ ] ε v [ ] Time [ ] Time [ ] 68

69 69 Conclusions Differential Algebra can improve efficiency of standard nonlinear methods improve accuracy of standard linear methods Note: method based on Taylor approximations Size of uncertainty set and order shall guarantee sufficient accuracy Polynomials can be manipulated to impose constraints on both initial and final state If linear methods are sufficiently accurate for your application, you may not need to increase order, however DA relieves you from the pain of writing variational equations

70 AstroNet-II International Final Conference Tossa de Mar June 2015 Relevant Applications of Differential Algebra in Astrodynamics Pierluigi Di Lizia in collaboration with: Roberto Armellin, Alessandro Morselli, Monica Valli, Alexander Wittig, Franco Bernelli Zazzera

71 71 Automatic Domain Splitting In challenging cases, a high-order Taylor expansion is not sufficient to accurately map uncertainties Initial set

72 72 Automatic Domain Splitting In challenging cases, a high-order Taylor expansion is not sufficient to accurately map uncertainties Initial set An algorithm can be implemented to automatically split the initial domain to guarantee the desired accuracy Very useful for: large uncertainty sets, long-term propagations, highly nonlinear dynamics

73 73 Sun t0 = 0

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79 79 Sun ti = 0.90 revs.

80 80 Sun ti = 0.96 revs. 3 boxes

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85 85 Sun ti = 1.91 revs. 7 boxes

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89 89 Computational time: 22 s (2.4 GHz Intel Core i5) Sun ti = 2.81 revs. 23 boxes

90 90 Automatic Domain Splitting Long term propagation of Apophis motion (NEODys in Sept. 2009)

91 91 Automatic Domain Splitting Long term propagation of Apophis motion weaker nonlinearity

92 92 BB1: Uncertainty Propagation Methods Nonlinear covariance propagation: Given the k-order polynomial map: Moments of the pdf can be obtained by computing the expectation of the Taylor expansion: for the mean computed by Isserlis s formula